Define an array m(i,j) by m(1,j)=m(j,1)=j*(j+1)/2 for j=0,1,2,3,... and m(i,j) = m(i,j-1) + m(i-1,j+1); the diagonal m(k,k) for k=1,2,3... gives the numbers in this sequence. - J. M. Bergot, May 02 2012

The central terms of triangle A051631 (including the root term -1) are given by (n-1)*(n+1)*Gamma(2*n+1)/Gamma(n+2)^2 with n>=0. - Peter Luschny, Nov 24 2013

Index the sequence from n=0 so that a(0)=1, a(1)=0,a(2)=2,a(3)=10,... a(n) is the number of walks using steps U=(1,1) and D=(1,-1) from the origin to (2n,0) that rise above and dip below the x axis. a(2) = 4 because we have: DUUD and UDDU. - Geoffrey Critzer, Jan 11 2014